Factorizations and Hardy-Rellich-Type Inequalities

Abstract

The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\alpha,\beta} := - \Delta + \alpha |x|^{-2} x \cdot \nabla + \beta |x|^{-2}$, $\alpha, \beta \in \mathbb{R}$, $x \in \mathbb{R}^n \backslash \{0\}$, $n \in \mathbb{N}$, $n \geq 2$, and its formal adjoint, denoted by $T_{\alpha,\beta}^+$, we show that nonnegativity of $T_{\alpha,\beta}^+ T_{\alpha,\beta}$ on $C_0^{\infty}(\mathbb{R}^n \backslash \{0\})$ implies the fundamental inequality, \begin{align} \int_{\mathbb{R}^n} [(\Delta f)(x)]^2 \, d^n x &\geq [(n - 4) \alpha - 2 \beta] \int_{\mathbb{R}^n} |x|^{-2} |(\nabla f)(x)|^2 \, d^n x \notag \\ & \quad - \alpha (\alpha - 4) \int_{\mathbb{R}^n} |x|^{-4} |x \cdot (\nabla f)(x)|^2 \, d^n x \notag \\ & \quad + \beta [(n - 4) (\alpha - 2) - \beta] \int_{\mathbb{R}^n} |x|^{-4} |f(x)|^2 \, d^n x, \notag \end{align} for $f \in C^{\infty}_0(\mathbb{R}^n \backslash \{0\})$. A particular choice of values for $\alpha$ and $\beta$ yields known Hardy-Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where $\mathbb{R}^n$ is replaced by an arbitrary open set $\Omega \subseteq \mathbb{R}^n$ for functions $f \in C^{\infty}_0(\Omega \backslash \{0\})$. Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order situations.